The number of leaves on a tree at
the start of the year 𝑛 is 𝐿 𝑛. The number of leaves on the tree at
the start of the following year can be found using the equation 𝐿 𝑛 plus one is
equal to 0.85 multiplied by 𝐿 𝑛 plus five. At the start of year one, there are
105 leaves on the tree. How many leaves will there be on
the tree at the start of year five? Give your answer to the nearest
We’ve been given an iterative
formula which tells us how to work out the number of leaves in the year 𝑛 plus one
from the number of leaves in the previous year, year 𝑛. It tells us that we take the number
of leaves at the start of the previous year 𝐿 𝑛, we add five to it, and then we
multiply it by 0.85. We’re told in the question that at
the start of year one there are 105 leaves on the tree, which means that the value
of 𝐿 one is 105.
To find the value of 𝐿 two, the
number of leaves at the start of year two, we substitute the value of 𝐿 one into
our iterative formula, giving 0.85 multiplied by 105 plus five. We can use a calculator to evaluate
this and it gives 93.5. To find the number of leaves at the
start of year three, we substitute our value of 𝐿 two into the iterative formula,
giving 0.85 multiplied by 93.5 plus five. This gives 83.725.
Now, just a tip here for when you
are using your calculator to evaluate these values, it’s a sensible idea to use the
answer button on your calculator. If you type what I’ve put on the
screen here, then your calculator would take the current value add five and then
multiply it by 0.85. This means that if you keep
pressing equals, your calculator will keep giving you the next value in this
We need to keep going because we
need to work out the number of leaves in years four and year five. I’ve written down the full
calculation for 𝐿 four. But in reality, I’ve just pressed
the equals button on my calculator. And it gives that 𝐿 four is equal
to 75.41625. Pressing equals one more time gives
the value of 𝐿 five 68.3538125.
The question asks us to give our
answer to the nearest whole number. And as the deciding number — that’s
the first digit after the decimal point — is a three, we round down. The number of leaves on the tree at
the start of year five is 68. Now, just a note about rounding,
I’ve kept the exact values on my calculator — so that’s the decimals — and then only
rounded the answer at the end.
You may decide instead to approach
this in a slightly different way. As the number of leaves on the tree
each year must be an integer; that is, a whole number, you may decide to round the
answers that you get for 𝐿 two, 𝐿 three, and so on.
The value of 𝐿 two was 93.5. So rounding this to the nearest
integer, this would give 94. Substituting 94 into the iterative
formula would give the value of 𝐿 three as 84.15. To the nearest whole number, this
is just 84. Substituting 84 into the iterative
formula would give the value of 𝐿 four as 75.65, which to the nearest integer is
76. Finally, substituting 76 into the
iterative formula would give the value of 𝐿 five as 68.85. This value rounds up to 69.
This is a slightly different answer
to the one we got when we only rounded at the end. But either method would be
considered acceptable here. The number of leaves on the tree at
the start of year five is either 68 or 69.